The current literature of randomized algorithms focuses on multivariate statistical methods which are not directly applicable to the functional data setting. Their theoretical results also does not naturally extend to functional data in an infinite dimensional space. [1]} recently studied randomized estimation of functional covariance operator. Still, most functional data methods remain to be explored via the randomized algorithm approach for scalability. Among these methods, functional principal component analysis (FPCA)
[2]}, [3]}, [4]} is one of the most important. Like the classical principal component analysis, FPCA identifies a few orthonormal functional principal components (FPC) explaining most of the variability of a dataset. The computation and theoretical analysis of functional data in an infinite-dimensional space are usually facilitated by FPCA. Indeed, a large amount of FDA methods have been proposed based on FPC. For example, when fitting functional linear regression, [5]}, [6]} first identify a few FPCs of the predictor, and then the model is fitted via the leading FPCs. For functional two-sample test, the test statistic can be constructed based on FPCs [7]}. The work of [8]} relies on FPCs to define probability density for random functions. More examples can be found in [9]}.
